## Topic outline

### Introduction

For several decades, gauge theory has been one of the driving forces of Geometry and Analysis. Its problems and methods have been foundational for many other fields. It is also an important topic in Physics.

This course will offer a brief introduction to the mathematical point of view on gauge theory and to some of its relationships to other parts of geometry. It should be of interest to students of both Geometry and Analysis.The total duration of the course will be 15 hours. It will attempt to cover the following topics. Some of these will be discussed only briefly and could be the topic of further self-study for those students who need to take the final exam.

- Smooth vector bundles, connections, curvature. The Yang-Mills functional.

- Flat bundles and connections. Holomorphic vector bundles.

- Overview of stability and of the Narasimhan-Seshadri theorem.

- ASD connections. Relationship to 4-dimensional topology.

- Moduli spaces of Yang-Mills connections. Compactness and bubbling phenomena.- The course will be in English, online.
PhD students needing academic credits for this course will be asked to deliver a lecture related to the course. Please contact me for details.Draft version. Do not trust, please let me know your corrections.

### Lecture 1: March 8, 17:00-18:30

Introduction to the course. Vector bundles with G-structures. Connections.

### Lecture 2: March 10, 17:00-18:30

Holonomy. Curvature. Yang-Mills functional.

### Lecture 3: March 15, 17:00-18:30

Gauge group. Flat connections, local systems.

### Lecture 4: March 17, 17:00-18:30

Monodromy representations and correspondence with flat connections/local systems. Complex manifolds, holomorphic bundles.

### Lecture 5: March 22, 17:00-18:30

Complex manifolds. Partial connections on complex vetor bundles. Integrability. Chern connections.

### Lecture 6: March 24, 17:00-18:30

Partial vs unitary connections: "Fix metric, find best connection. Fix connection, find best metric". Examples for line bundles over Riemann surfaces.

### Lecture 7: March 29, 17:00-18:30

Slope and stability of holomorphic vector bundles. HYM connections. Narasimhan-Seshadri and DUY theorems.

### Lecture 8: March 31, 17:00-18:30

Algebraic topology, differential topology, geometry: 3 viewpoints on cohomology.### Lecture 9: April 5, 17:00-18:30

Instantons. Topological lower bounds and minimizers in gauge theory and calibrated geometry.

### Special lecture: April 7, 17:00-18:30

Fabio Paradiso (University of Torino) will present the following topic.

**Title:**Donaldson's Diagonalization Theorem.

**Abstract:**The Diagonalization Theorem is one of the main results which earned Simon Donaldson the Fields Medal in 1986. The theorem states that, when definite, the intersection form of a compact, oriented, simply-connected smooth 4-fold is diagonalizable. Its importance lies in the great insight it provides about the interplay between the smooth and topological categories in four dimensions. In particular, it can be used to prove the existence of exotic smooth structures on the 4-dimensional euclidean space.

In this talk, we shall examine a sketch of the proof of Donaldson's Diagonalization Theorem, which involves the construction of the moduli space of anti-self-dual connections on SU(2)-bundles over 4-folds.### Special lecture: April 12, 17:00-18:30

Marco Golla (University of Nantes) will present the following topic (abstract below).

----------------------------------------------

Topic: Gauge theory — guest lecture

Time: Apr 12, 2022 05:45 PM Paris

Join Zoom Meeting

https://univ-nantes-fr.zoom.us/j/89267558152?pwd=dzNXeG9DOUZ 5SFozM3l5NitTeTJnUT09

Meeting ID: 892 6755 8152

Passcode: 011235813-----------------------------------------------

**Title:**Some applications of Donaldson's diagonalisation theorem

**Abstract:**Donaldson's diagonalisation theorem has had a huge impact on

the study of smooth 4-manifolds, especially in combination with

Freedman's counterpart results on topological 4-manifolds. We will

examine some applications to exotic 4-manifolds, embedded surfaces in

4-manifolds, knots, homology cobordisms of 3-manifolds, and embeddings

of 3-manifolds in 4-manifolds.- This topic
### Lecture 2: 16/11/2020, 16:00-18:00

### Topic 23